Quantum loss sensing with two-mode squeezed vacuum state under noisy and lossy environment

We investigate quantum advantages in loss sensing when the two-mode squeezed vacuum state is used as a probe. Following an experimental demonstration in PRX 4, 011049, we consider a quantum scheme in which the signal mode is passed through the target and a thermal noise is introduced to the idler mode before they are measured. We consider two detection strategies of practical relevance: coincidence-counting and intensity-difference measurement, which are widely used in quantum sensing and imaging experiments. By computing the signal-to-noise ratio, we verify that quantum advantages persist even under strong thermal background noise, in comparison with the classical scheme which uses a single-mode coherent state that directly suffers from the thermal noise. Such robustness comes from the fact that the signal mode suffers from the thermal noise in the classical scheme, while in the quantum scheme, the idler mode does. For a fairer comparison, we further investigate a different setup in which the thermal noise is introduced to the signal mode in the quantum schemes. In this new setup, we show that the quantum advantages are significantly reduced. Remarkably, however, under an optimum measurement scheme associated with the quantum Fisher information, we show that the two-mode squeezed vacuum state does exhibit a quantum advantage over the entire range of the environmental noise and loss. We expect this work to serve as a guide for experimental demonstrations of quantum advantages in loss parameter sensing, which is subject to lossy and noisy environment.

The quantum Fisher information (QFI) can be calculated from the fidelity F(ρ T , ρ T +dT ) between the two states ρ T and ρ T +dT and is given by [1] There is a well-known method of computing the fidelity of the Gaussian states [2]. For n bosonic modes described by quadrature operators Q = (x 1 , · · · , x n , p 1 , · · · , p n ) T , the canonical commutation relations can be written as [3] [Q, Q T ] = iΩ n , Ω n := 0 n I n −I n 0 n where I n and 0 n are the n × n identity matrix and null matrix, respectively. The first-and second-order moments of Gaussian states can be written in terms of the mean (u) and the covariance matrix (V ) as follows: Using a modified version of the covariance matrix W := −2V iΩ n , the fidelity between two Gaussian states is given by where δ u := u 2 − u 1 and F 0 (V 1 , V 2 ) is given by [2,4,5].
for single-mode Gaussian states and for two-mode Gaussian states. Here, the symplectic invariants are defined as ∆ = det(V 1 +V 2 ), Γ = 2 2n det(Ω n V 1 Ω n V 2 − I/4), and Λ = 2 2n det(V 1 + iΩ n /2)det(V 2 + iΩ n 2). The first-and second-order moments of the classical and quantum schemes are given as where a and b are the complex components of the displacement parameter α = √ n. Note that n here denotes the mean signal photon number. Now, one can compute the quantum Fisher information of the classical and quantum schemes using Eqs. (1) ∼ (9).
A. Outline of the calculation procedure for QFI Let us start with the classical scheme for the original setup. To calculate the QFI, we use Eq. (5) to calculate the fidelity first, then use Eq. (1). Wtih ρ 1 = ρ T and ρ 2 = ρ T +dT , the first-and second-order moments are readily obtained as Equation (6) gives F 0 (V 1 , V 2 ) = 1 and from Eq. (5) we obtain To obtain the QFI, we first Talyor expand Eq. (12) about dT to obtain and substitute it into Eq. (1). This yields the QFI of the classical scheme for the original setup: . The procedure is more or less the same in the quantum scheme, but with different first and second-order moments. We also need to use Eq. (7) instead of Eq. (6) because we are dealing with a two-mode state. Because the first order moment u Q is zero the exponential term in Eq. (5) is 1, and all we need is the F 0 term. The latter is too cumbersome to write down, but nevertheless calculable with the aid of a symbolic programming language form which the QFI can be calculated in the same way as above. The QFIs for the alternative setup are calculated in the same fashion, but with different density operator ρ T , hence different first and second-order moments.

II. UNBIASED ESTIMATORS AND VARIANCES
Here we outline the procedures for (i) obtaining the unbiased estimators used in the main text and (ii) calculations of variances. Because the procedure is the same for the original and alternative setups, we will only provide outlines for the original setup.

A. Classical scheme
The initial state of the classical scheme is ρ in = ρ coh ⊗ ρ vac ⊗ ρ n ′ th , where ρ coh , ρ vac , and ρ n ′ th represent the input coherent, vacuum, and thermal states, respectively. The bosonic operator of the output state is given by where a in , c in , and d in are the annihilation operators of the input coherent, vacuum, and thermal states, respectively and T and η are the transmittances of the sample and the noisy environment, respectively. To obtain an unbiased estimator, we calculate the expectation value for the photon number counting measurement: From which one readily obtains an unbiased estimatorT C = a † out aout−n th ηn . The variance of the unbiased estimator is then T (2n th + 1) + n th (n th +1) ηn ηn . (17)

B. Quantum schemes
In the quantum case, the initial state of the quantum scheme is changed to ρ in = ρ tmsv ⊗ ρ vac ⊗ ρ n ′ th , where ρ tmsv , ρ vac , and ρ n ′ th represent the input two-mode squeezed vacuum state, the vacuum (for the sample), and thermal states, respectively. Annihilation operators of the output modes are given by where a in , b in are the annihilation operators of the two-mode squeezed vacuum state and c in , and d in are the annihilation operators of the vacuum state and thermal state, respectively. For the photon number difference measurement, the average is from which an unbiased estimator is readily obtained as,T ND = (a † out aout−b † out bout)+ηn+n th n . The variance of the unbiased estimator is then The operator for the coincidence counting scheme is given by a † out a out b † out b out . The expectation value is which yields an unbiased estimatorT The variance of the unbiased estimator is T n (η(2n 2 + n) + nn th ) 2 n 2 th (3T n + 2) + n th [1 + 4η + 20T ηn 2 + 2n(T + 4η + 7T η)] + η[1 + 20T ηn 3 + n(2 + 4T + 4η + 3T η) + n 2 (6T + 6η + 20T η)] .
III. QUANTUM ENHANCEMENTS FOR n th = 0 Figures S1(a) and (b) depict how Figs. 3 and 5 in the main text change, respectively, in the limit of vanishing thermal photon number, i.e., n th → 0. For the original asymmetric setup (Fig. S1(a)), the 'No enhancement' region occupies a larger parameter space at the cost of reduced region for 'coin', while the region occupied by 'diff' stays more or less the same. The values of R coin have decreased significantly from those at n th = 0.1, while R diff exhibits mixed behavior. The latter has increased when T is large and γ is small (bottom right corner), but has decreased when γ is large (for all T ). The situation is similar for the alternative quantum setup, as shown in Fig. S1(b), except for the fact that the coincidence-counting scheme exhibits no quantum enhancement at all when n th = 0.

IV. PLOTS OF ∆T
This section provides plots of the standard deviations of the estimator, i.e. ∆T , achieved by the quantum and classical schemes. Figures S2 and S3 are plots of ∆T in the original and alternative setups, respectively. Note that the results are for a single copy of the input state. For N copies, the standard deviation is divided by √ N .